|Thursday, March 22||Amie Wilkinson, University of Chicago|
|2:30pm||The first digit|
|ABSTRACT||Write down a list of numbers and look at their first digits. Is there a pattern? I will discuss methods from dynamical systems that can be used to understand the first digits of numbers encountered in everyday life.|
|Thursday, April 19||Vladimir Retakh, Rutgers University|
|2:30pm||Commutative and Noncommutative Symmetric Functions|
|ABSTRACT||The classical theory of commutative symmetric functions started a long
time ago when people realized the connection between coefficients of
polynomial equations and their roots. Nowadays, symmetric functions
appear in almost every area of mathematics and the theory continues to
grow. The theory of noncommutative symmetric functions is much younger but but
we can already count a number of interesting results and applications in
this direction. The talk is dedicated to basic notions and connections of the both
theories of symmetric functions.
|Thursday, April 5||John Roe, The Pennsylvania State University|
|2:30pm||Being right more often can make you more wrong|
|ABSTRACT||Suppose that we know the values of a function f(x) at (n+1) different x-values, say in the interval [0,1]. A polynomial function p(x), of degree n, can be put through these (n+1) points and it is natural to use p as an approximation to f for other x-values also. This idea, called "polynomial collocation", is one of the oldest in numerical analysis.
I'll explain some of the techniques that were used in the B.C. (=before computer) era to carry out these calculations using pencil and paper. Then, I'll show you a surprising fact: as n increases (so that the number of points at which the approximation is "right" gets larger), the difference between p(x) and f(x) for other x-values can increase without bound. In brief, being right more often can make you more wrong.
|Thursday, January 19||Mark Levi, The Pennsylvania State University|
|2:30pm||Physical discoveries and proofs of mathematical theorems|
|ABSTRACT||Physics often provides mathematics not only with a problem, but also with the idea of a solution.
Some calculus problems can be solved more quickly without calculus, by using physics instead.
Quite a few theorems which may seem somewhat mysterious become completely obvious when given a proper physical incarnation.
This is the case for some “elementary” theorems (the Pythagorean Theorem, Pappus' theorems, some trig identities, Euler's formula V-E+F=2, and more)
and for some less elementary ones: Green's theorem, the Riemann Mapping Theorem, Noether's theorem on conserved quantities,
and more (no familiarity with any of these is assumed). I will describe a miscellaneous sampling of problems according to the audience's preferences.
|Thursday, February 9||Jana Rodriguez Hertz, IMERL, Uruguay|
|2:30pm||Lake fish population estimates and mixing properties|
|ABSTRACT||A mark-and-recapture method of lake fish population estimate consists in collecting and tagging a random sample of, for instance, 1000 fish, and counting the tagged fish in a second random capture, after releasing and allowing a period of mixing. A number of, for instance, 10 tagged fish would give an estimate of 100,000 fish in the lake. Is this a good estimate? We will see how this depends on the way the fish get mixed.|
|Thursday, February 23||Jan Reimann, The Pennsylvania State University|
|2:30pm||The Continuum Hypothesis|
|ABSTRACT||In the late 19th century the German mathematician Georg Cantor tried
to show that every uncountable subset of the real numbers can be
mapped bijectively onto the real line. He was unable to prove this,
and the question became the Continuum Hypothesis (CH). It was the
first question on Hilbert's famous problem list of 1900. Seminal works
by Gödel and Cohen showed that CH can neither be proved nor disproved
from Zermelo-Fraenkel set theory (ZF), a basic axiom system for sets
that captures most of modern mathematics. In other words, CH is
independent of ZF.
In this talk I will sketch the history of the Continuum Hypothesis,
how it influenced the development of logic and set theory in the 20th
century, and I will outline how one can show that a statement is
independent of ZF.
|Thursday, March 15||Eugene Wayne, Boston University|
|2:30pm||The Navier-Stokes equations|
|ABSTRACT||We are surrounded by fluids in motion. As a consequence, they
were among the first physical systems that physicists and applied
mathematicians tried to model and the currently accepted mathematical
model for fluid motion, the Navier-Stokes equations, are more than 150
years old. As a consequence, one might assume that all the interesting
mathematical questions about these equations have been answered. However,
when the Clay Mathematics Foundation made a list of seven "Millennium
Prize Problems" (with a one million dollar prize for their solution),
one of the seven was to show that the three-dimensional Navier-Stokes
equations has a smooth solution. I will explain the physical origin
of these equations, some of the surprising phenomena that appear
in these equations, and why they are so hard to solve.